d. Planning, strategies and metacognitive aspects of analogical reasoning

Planning, strategy implementation and metacognitive aspects are fundamentally linked to executive functions, and are involved in strategy use: they permit to act on the substratum of the task to pursue goals which are ordered in time. Thus differences in strategies are likely to appear with the maturation of executive functions. Planning and goal management are also tightly related to subjects' understanding of the task and its constraints on the solution, because this understanding will lead to a strategy rather than another.

Mature reasoning

One of the first studies that investigated the different strategies used to solve analogical problems is the study of Whitely & Barnes (1979) which used protocol analysis in participants' requests for information when solving verbal A:B::C:? problems using unreal animals. Because the labels did not correspond to any known animal, participants had to request information from the experimenter to correctly solve the analogies. The full set of animals' properties was given to the participants at the same time. Overall Whitely & Barnes found an important between-subject variability in the strategies: 18% solved the problem

89 directly after encoding, 23% made one confirmation step before answering, 45% requested information about the whole stem (i.e., A, B and C) before searching for a solution, only 10%

asked more information about C before searching for a solution. Other strategies were only marginally observed. This suggests individual differences between participants in their ability to deal with the amount of information given to them (supported by executive functions and working memory capacity) in the task, and to articulate it in a meaningful analogy, leading to different strategies.

Bethell-Fox, Lohman, & Snow (1984) were the first to address the question of participants' strategies in analogical reasoning recording eye-movements data. They hypothesized a strategy shift within subjects in simple and difficult geometrical A:B::C:?

trials, and a difference of strategy between high fluid intelligence and low fluid intelligence participants. They found evidence for two different strategies: constructive matching, used in easier problems, and more widely by participants with high fluid intelligence, and response elimination, used more in harder problems and by participants with low fluid intelligence.

Constructive matching consists in constructing a full model for the answer on the basis of the first three terms of the analogy, and response elimination consists in eliminating the answers that do not fit on the basis of partial evidence from the stem of the analogy.

Mulholland et al. (1980) also explored their participants’ strategies in a geometrical analogy judgment task. In this task, participants first encoded exhaustively the difference between A and B, then generated the rules for changing A into B, which was followed by a phase of encoding C and D, mapping of the attributes of A:B on C and D, and comparing the different operations transforming C into D to those stored in working memory between A and B. A answer is given if one of the operations is not present or is not the same as in A and B or if all inferred operations correspond to those between A and B. This strategy, different from the one postulated by Sternberg (1977; see above) and Whitely & Barnes (1979), is most probably the resultant of the task constraints that differ from analogical judgment (in which a D term is given and has to be evaluated) to analogical problem solving (in which the D term has to be found between different solution options).

These studies show that adults can adapt their strategies in response to the limitations of their executive functions and working memory. However these different strategies lead to correct answer most of the time, which suggests that they understand correctly the constraints on the solution of the task, and that these strategies are adapted to these constraints.

90 Reasoning development

However, children's knowledge about the constraints of the task and their own limitations, and their ability to manage the different goals of analogical reasoning tasks seem to be limited at first, and to develop over the years, as the following review demonstrates.

One of the earliest positions about children's strategies development in analogical reasoning task is a shift from associative to analogical responding due to maturation processes (Piaget et al., 1977) and the extent to which children used associative responding was negatively correlated with their IQ and achievement (Achenbach, 1970a, 1970b), even in the long run (Achenbach, 1971). Associative errors were also negatively correlated with working memory capacity (measured by a simple digit span task) and semantic flexibility (measured as the ability to find in a list of words the different meanings of a word when in the context of words coherent with only one meaning) in 10-to-11-year-olds (Tirre, 1983).

Sternberg & Nigro (1980) observed in their children (9-, 12-, 15-, and 18-year-olds) an increase in consistency in the use of strategies across trial difficulties, and found differences in the models accounting for children's performances with age. First, children's (9 and 12-year-olds) reasoning tend to be guided by the association strength between the last word of the stem of the analogical problem and the first word of the solution option, suggesting an incomplete form of analogical reasoning, with an incomplete encoding of the analogical problems (possibly linked to a limitation in working memory). Older children and adults (15 and 18-year-olds) rely more exclusively on their reasoning abilities to relate the analogical problem parts and tend to have exhaustive processes, suggesting less limitation from working memory capacity in their strategy. Similar reliance on association was observed in older children and adults when learning novel domains (Alexander, Murphy, & Kulikowich, 1998).

Even though most of the materials in the preceding studies were semantic, a similar trend toward using associated matches were found in geometrical analogies (Alexander, Willson, White, & Fuqua, 1987). In this study, children (4-to-5 years of age) who did not solve the problems by analogy relied on a hierarchical set of rules to choose their answer: they used the similarity (which could be interpreted as perceptual association) between the potential answers and the C terms of the problems. Thus, non analogical reasoners tended to

91 choose the exact match with C, or, if absent from the solution set, an item that was identical to the greatest number of dimensions possible, instead of the correct answer.

The same bimodality of responding (i.e., similarity or analogical matching) between the age of 5 and 11 has been found in an analysis of several sets of data using geometrical analogies varying the relational complexity of the transformation from A to B, and was interpreted as an indicator of a developmental discontinuity in the way children apprehend analogical problem solving (Hosenfeld, van der Maas, & van den Boom, 1997b). This transition, indicated by bimodality between subjects of different age, was also shown in longitudinal data (Hosenfeld et al., 1997a). Six-to-eight-year-old children were tested in the same geometrical analogy task as in Hosenfeld et al. (1997b) with eight sessions on a six-month period. They identified four indicators of transition between two modes of responding:

bimodality (i.e., two different strategies of responding between subjects), sudden jump (i.e., sudden change of responding within subject), anomalous variance (i.e., increase in variance due to a conflict between the two modes of responding near the transition point), and critical slowing down (i.e., increase in reaction time due to the conflicting strategies near the transition point). Convergent findings indicating a developmental transition around the same age was found in simple 2x2 Raven Progressive Matrices task which is another test of relational reasoning close to the A:B::C:? task (Siegler & Svetina, 2002).

Interestingly, a similar trend from associative to analogical reasoning has been observed in adolescents in more complex, third order analogies (i.e., analogies between analogies; (Sternberg & Downing, 1982). Samples of 13-, 16-, and 18-year-olds were tested in a meta-analogical judgment task. They were presented with two analogies and asked to rate on a 1-to-9 scale how analogically related were these analogies (e.g., how analogically related were sand:beach::star:galaxy and water:ocean::air:sky). The authors also showed that the ability to map one domain on the other (i.e., to observe the similarity between the relations in the two domains) was the last to develop in this meta-analogy task, as it was in younger children in more simple analogy tasks (Sternberg & Rifkin, 1979). This development of the ability to map analogies one on the other and thus extract more complex schemas could lead to meta-analogical skills (i.e., the ability to draw analogies about analogies, Burns, 1996). The observation of the same development from associative to analogical reasoning at two different ages with problems of increasing complexity suggests that the ability to make chunks of increasing information is crucial for the ability to judge and use similarity between domains.

92 Hence, during development, an increase in children's ability to take into account the main goal of analogical reasoning (i.e., to compare the two domains on the basis of their similarity) and in their understanding of how it is implemented could explain the decrease in associative responding. Children's comprehension of the task itself and its constraints was shown to be decisive in the ability to solve analogical problems (Goswami et al., 1998). In this study, children aged 4 to 5 were tested in A:B::C:? problems using causal transformations between both pairs of pictures, and controlled in their knowledge of the relations used. They varied the number of relations (either one or two) and the order of the different tasks (single relation A:B::C:? task, causal reasoning control task, double relation A:B::C:? task). They found a significant effect of the number of relations, mostly due to the poor performances in the double relation analogical reasoning task when it was the first task. However, children presented first with the single analogy task did not have different performances between the single and double relation A:B::C:? task. These results can be explained by the fact that a first exposure to the task with simple relations helps them understand the constraints put on responding in the A:B::C:? task, even in more complex trials. Therefore, young children's associative responding might be explained by their inability to understand the structure of the task itself or to keep it active, especially with difficult stimuli. Training them with a simpler task might consolidate their representation of the constraints on the solution of the problems.

These results suggest also that the attentional focus on relation observed in children who are trained with a progressive alignment procedure might attend more to relational features because of a better understanding of the task constraints.

Other evidence showing children's poor understanding of the task constraints have been gathered (Cheshire, Ball, & Lewis, 2005). In this study, children (aged 6 to 7) were tested over several sessions in different conditions (control, practice only, feedback, self-explanation of the answer, and self-self-explanation and feedback) in a simple 2x2 Raven Progressive Matrices test. They showed that children both benefitted from the self-explanation and the feedback condition. However, children in the self-self-explanation and feedback benefited from an additive effect between those two helps. The feedback condition was argued to have an effect on children's representation of the constraints of the task on the adequate solution, when the self-explanation condition was interpreted as helping children maintaining attention toward the goals of the task. This explanation is plausible as children have been shown to demonstrate goal neglect in complex tasks (Blaye & Chevalier, 2011;

Chevalier & Blaye, 2008a; Marcovitch et al., 2010). Thus, children's failure in analogical

93 reasoning task could be due at least in part to a failure to represent correctly the task's goals and keep them active throughout their search.

Many studies involved children's training in analogical reasoning task. Alexander and colleagues (Alexander et al., 1989; Alexander, Haensly, Crimmins-Jeanes, & White, 1986;

White & Alexander, 1986; see also Tunteler, Pronk, & Resing, 2008; Stevenson, Heiser, & M.

Resing, 2013) trained children in the different cognitive components of analogical reasoning problems (i.e., encoding, inference, mapping, and application; Sternberg, 1977). They found improvement in the children observed both in verbal and geometrical A:B::C:? tasks, suggesting that a better understanding of the tasks' constraints and goals positively affects children's performance.

Clues indicating a possible goal neglect in children were observed in children's eye-movements gathered while they were solving a semantic A:B::C:? task (Thibaut, French, Missault, et al., 2011). The authors compared children's (5- and 8-year-olds), adolescents' (14-year-olds) and adults' visual strategies and found that both children's groups were less likely to gather information from A and B than adolescents and adults. This indicates a potential neglect of a major subgoal of the task (i.e., finding the AB relation in order to compare it to the possible solution's relations to C), which might cause at the basis of children's associative strategy: they might focus on the goal of the task that is the most emphasized (i.e., finding something that goes with C) without taking into account the constraint that this solution should be linked to C in the same way as B is linked to A.